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∈ SO(n), (t ∈ R)
In fact, any Lie group G has a Lie algebra g and there is a map called exponentiation
exp: g → G
such that exp((s + t)A) = exp(sA) exp(tA).
For example, if g = R (a vector spce) and G = R+^ (the Lie group of positive real numbers, with multiplication as the group operation), then
exp : R → R+
is the usual exponential function and I am just saying exp(s + t) = exp(s) exp(t). Here a slide rule (or table of logarithms) converts problems in the Lie group into problems in the Lie algebra.
Definition 1 A Lie algebra is a (real) vector space g equipped with an operation
[·, ·]: g × g → g
called the Lie bracket which has the following properties:
- bilinearity
- antisymmetry
- the Jacobi identity
We’ve seen one type of Lie algebra so far: if X is a Poisson manifold, C∞(X) is a Lie algebra with the Poisson bracket as its bracket. But in fact, every manifold gives a Lie algebra, in a different way:
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Theorem 2 Given any manifold X, let Vect(X) be the set of vector fields on X. This becomes a Lie algebra by: (αv)f = α(v(f )) (v + w)f = vf + wf [v, w]f = v(wf ) − w(vf )
where α ∈ R and f ∈ C∞(X).
Sketch of proof: Just calculate to check the Lie algebra axioms. For example: a vector field v ∈ Vect(X) is a linear map v: C∞(X) → C∞(X)
satisfying v(f g) = v(f )g + f v(g).
Why does v, w ∈ Vect(X) ⇒ [v, w] ∈ Vect(X)?
[v, w](f g) = vw(f g) − wv(f g)
= v(w(f )g + f w(g)) − w(v(f )g + f v(g)) = (vw(f ))g + w(f )v(g) + v(f )w(g) + f (vw(g)) − wv(f )g − v(f )w(g) − w(f )v(g) − f (wv(g)) = ([v, w]f )g + f ([v, w]g).
Why does [·, ·] satisfy the Jacobi identity?
[u, [v, w]] = [[u, v], w] + [v, [u, w]]
([[u, v], w] + [v, [u, w]])f = (uvw − vuw − wuv + wvu + vuw − vwu − uwv + wuv)f = [u, [v, w]]f
Etc....
This Lie algebra Vect(X) is very related to the Poisson algebra C∞(X) when X is a Poisson manifold:
Theorem 3 If X is a Poisson manifold and f, g ∈ C∞(X) then
[vf , vg ] = v{f,g}.
Proof - For any h ∈ C∞(X),
v{f,g}h = {{f, g}, h} = {f {g, h}} + {{f, h}, g} = {f, {g, h}} − {g, {f, h}} = vf vg h − vg vf h = [vf , vg ]h
Later we’ll define a ‘homomorphism’ between Lie algebras, and the theorem we just proved will say that v 7 → vf
is a homomorphism from the Lie algebra C∞(X) to the Lie algebra Vect(X).
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